2 Answers
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In summary, if you're close to the North (or South) Pole and it's dark for days, the opinions are:

Minchas Elazar: halachically, night can last for several weeks or months. So don't put on tefilin if it's dark.

Tiferes Yisroel: bring along an almanac from your hometown, and follow that. Absent normal observable astronomy, you pretend you're back in New York or wherever.

Ben Ish Chai: absent normal observable astronomy, revert to a day that is 6AM-6PM; with dawn and twilight periods each of 72 minutes. (If you come from Ecuador and follow the Tiferes Yisroel, this is basically what you'd be doing anyhow.)

Moadim Uzmanim: use midnight and noon as your dividing lines.

If you're ever stuck in a situation like this for real, ask your rabbi which of these opinions (or combination thereof) to follow! See the article for Rabbi Heinemann's psak combining some of these.

With regards to space travel, I recall hearing another opinion that you revert to Jerusalem time, but I don't recall the source.

I think the Chabad rabbi in Florida told Ilan Ramon Z"L to use Florida time, since that's where he'd been living before he took off.
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Isaac MosesJan 13 '11 at 15:08

Can you rell us Rabbi Heinemann's opinion
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SimchasTorahJan 13 '11 at 23:46

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Rabbi Heinemann accounts for all of the above opinions: when it's totally dark all winter, Shabbos starts at noon on Friday (Moadim Uzmanim), and ends at 7:12PM (Ben Ish Chai). Morning davening can be said when it's morning in your hometown (Tiferes Yisroel), but no earlier than 6AM (Ben Ish Chai). But when you daven, we're not sure you're obligated (Minchas Elazar), so don't say the brachos on Shema, and have in mind your Shemoneh Esrei may be voluntary. Phew!
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ShalomJan 14 '11 at 13:33

I recollect that friend of mine told me that his father-in-law was stationed in Greenland and was told by R' Moshe (or by someone who asked for him?) that he should go by the closest city with normal times.

I haven't done a whole lot of research on this subject, but off the cuff, perhaps all of these times aren't "real times", but a personal gauge. According to the Maharsha 26b, even Rabbi Yose b' R' Chanina, who holds tefilos keneged avos tiknum, also held of k'neged korbanos tiknun, but Chazal would not have established the tefilos at those korbanos times if not for the original tefilos established by the Avos who fulfilled the pasuk of erev vavoker v'tzaharayim asicha... So it could be that absent the times of chazal, the halacha would revert to 3x/day even without hard zemanim. Although the pasuk says specifically evening, morning and afternoon, it's possible to say that's lav davka as Tosfos proves that Yaakov davened maariv during the day (for Tosfos a proof that the halacha is like R' Yehuda). Lulei mistafina midivrei Tosfos, I would have said the timing mentioned in the pasuk is lav davka (and this is how the rabanan who argue w/ R"Y would answer Tosfos' question).

Krias Shema is a little fuzzier. Are the zemanim of 3 hours, etc. mideoraysa, or applications of Uvshochb'cha uvkumecha. In other words, can we say in absence of zemanim, we revert to shechiva and kima of your time zone? It seems acc. to the opinions above that that wouldn't be the case.

"go by the closest city with normal times." Here's what Wikipedia says about the South Pole Station: "The place has no solar time; there is no daily maximum or minimum solar height above the horizon. The station uses New Zealand time (UTC+12, UTC+13 during daylight saving time) since all flights to McMurdo station depart from Christchurch and therefore all official travel from the pole goes through New Zealand."
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ShalomJan 19 '11 at 15:31